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Method and apparatus for constructing a perfect trough parabolic reflector

Abstract

A coordinate system defines the length of the curve of a parabola used in
constructing a parabolic trough reflector. The origin (0,0) of the
coordinate system is at the bottom center of the coordinate system. The
two upper points of the coordinate system define the width, height of the
parabola. These points are defined as (X1,Y1)=(-width,height), and
(X2,Y2)=(width,height). The equation defining the parabola is
f(x)=a.multidot.x.sup.2,where a=height/width.sup.2. The plot of this
equation produces a parabola that fits into the coordinate system. Two
small blocks are used as anchor points for the ends of the parabola. The
length of the curve of the parabola is defined in the equation:
length(x)=a.multidot.[x.multidot.(x{square root}{square root over
(x.sup.2+b.sup.2)})+b.sup.2.multidot.ln(x+{square root}{square root over
(x.sup.2+b.sup.2)})]
where b=1/2.multidot.a.
An inexpensive trough reflector is constructed out of flexible material.
It is used to build a much more complicated six reflector system to
concentrate parallel radiation like sunlight much like a magnifying
glass. This system also forms the basis for building a much more powerful
telescope.

1. A method for constructing a parabolic trough reflector, comprising the
steps of: defining a coordinate system; calculating the dimension of,
including the length of the curve of the parabola, and making a parabolic
trough reflector that fits into the coordinate system at three specific
intercept points; forming a support structure as defined by the
coordinate system; and securing the parabolic trough reflector in the
support structure where each end of the length of the parabola trough is
secured in a mathematically precise slot in the support structure, these
slots are required to have a slope of the 1.sup.st derivatives of the
parabola at the points where the parabolas intercept the slots.

2. The method according to claim 1, wherein the length of the curve of the
parabola is defined as: f(x)=a.multidot.x.sup.2b=1/2.multidot.a
length(x)=a.multidot.[x.multidot.({square root}{square root over
(x.sup.2+b.sup.2)})+b.sup.2.multidot.ln(x+{square root}{square root over
(x.sup.2+b.sup.2)})]and; the Length of the curve=length (X2)-length (X1).

3. The method according to claim 1, wherein the securing of the parabolic
trough reflector defined in the coordinate system is done on at least
three points along the length of the curve of the parabola.

4. A method for constructing a parabolic trough reflector, comprising the
steps of: defining a coordinate system; calculating the dimension of,
including the length of the curve of the parabola, and making a parabolic
trough reflector that fits into the coordinate system at three specific
intercept points; forming a support structure as defined by the
coordinate system; and securing the parabolic trough reflector in the
support structure on at least three points along the length of the curve
of the parabola defined by the origin and two end points of the parabola.

5. A parabolic trough reflector, comprising: a coordinate system defining
the length of the curve of the parabola; a parabolic trough reflector
mounted in a rigid support structure as defined by the coordinate system,
each end of the reflector having an extended length secured in a mounting
block with defined slots.

6. The parabolic trough reflector according to claim 5, wherein the
parabolic trough reflector is secured to the support structure on at
least three points along the length of the curve of the parabola using a
mathematically precise anchoring system.

7. The parabolic trough reflector according to claim 5, wherein the
coordinate system includes a structure for mounting the parabolic trough
reflector.

8. The parabolic trough according to claim 5, wherein the parabolic trough
system is a two reflector system that is aligned to concentrate parallel
radiation in a horizontal or vertical direction.

9. The parabolic trough according to claim 5, wherein the parabolic trough
system is a six reflector system that is aligned to concentrate a
rectangular aperture of parallel radiation to a much smaller aperture of
parallel radiation.

10. The parabolic trough according to claim 9, wherein the six reflector
parabolic trough system includes two sets of three reflectors.

11. The parabolic trough according to claim 9, wherein the parabolic
trough system utilizes a six reflector system to form a telescope that
has the reflectors aligned to a smaller telescope to concentrate a small
rectangular aperture of parallel radiation to a focal plane.

[0002] The invention relates to parabolic reflectors and more particularly
to a method and apparatus for constructing multiple parabolic reflectors
from flexible material using a mathematically precise clamping system for
various purposes.

BACKGROUND OF THE INVENTION

[0003] Parabolic reflectors can be constructed by shaping a flexible
material to the parabolic shape. This is accomplished by bending the
flexible material to form a parabola. In some instances, the parabola may
be formed by molding the material to the parabolic shape and coating it
with a suitable material.

[0004] In U.S. Pat. No. 4,115,177, a tool is provided for manufacturing
parabolic solar reflectors. The tool employs an improved smooth convex
parabolic surface terminating in edges remote from the parabolic vertex
which are preferably placed under elastic tension tending to draw the
edges toward each other. The improved convex surface is a film of plastic
coated with chromium metal on its exterior surface. A multiple layered
thermosetting plastic reflector support is molded onto the convex surface
of the tool. The reflector support is removed from the tool and a layer
of aluminum is vacuum deposited onto the interior concave parabolic
reflector surface.

[0005] In U.S. Pat. No. 4,571,812, a solar concentrator of substantially
parabolic shape is formed by preforming a sheet of highly reflective
material into an arcuate section having opposed longitudinal edges and
having a predetermined radius of curvature and applying a force to at
least one of the opposed edges of the section to move the edges toward
each other and into a predetermined substantially parabolic configuration
and then supporting it.

[0006] A parabolic trough solar collector using reflective flexible
materials is disclosed in U.S. Pat. No. 4,493,313. A parabolic cylinder
mirror is formed by stretching a flexible reflecting material between two
parabolic end formers. The formers are held in place by a spreader bar.
The resulting mirror is made to track the sun, focusing the sun's rays on
a receiver tube. The ends of the reflective material are attached by glue
or other suitable means to attachment straps. The flexible mirror is then
attached to the formers. The attachment straps are mounted in brackets
and tensioned by tightening associated nuts on the ends of the attachment
straps. This serves both to stretch the flexible material orthogonal to
the receiver tube and to hold the flexible material on the formers. The
flexible mirror is stretched in the direction of the receiver tube by
adjusting tensioning nuts. If materials with matching coefficients of
expansion for temperature and humidity have been chosen, for example,
aluminum foil for the flexible mirror and aluminum for the spreader bar,
the mirror will stay in adjustment through temperature and humidity
excursions. With dissimilar materials, e.g., aluminized mylar or other
polymeric material and steel, spacers can be replaced with springs to
maintain proper adjustment. The spreader bar cross section is chosen to
be in the optic shadow of the receiver tube when tracking and not to
intercept rays of the sun that would otherwise reach the receiver tube.
This invention can also be used to make non-parabolic mirrors for other
apparatus and applications.

[0007] In U.S. Pat. No. 4,348,798, an extended width parabolic trough
solar collector is supported from pylons. A collector is formed from a
center module and two wing modules are joined together along abutting
edges by connecting means. A stressed skin monocoque construction is used
for each of the modules.

[0008] In U.S. Pat. No. 4,135,493, a parabolic trough solar energy
collector including an elongated support with a plurality of ribs secured
thereto and extending outwardly therefrom. One surface of said ribs is
shaped to define a parabola and is adapted to receive and support a thin
reflecting sheet which forms a parabolic trough reflecting surface. One
or more of said collectors are adapted to be joined end to end and
supported for joint rotation to track the sun. A common drive system
rotates the reflectors to track the sun; the reflector concentrates and
focuses the energy along a focal line. The fluid to be heated is
presented along the focal line in a suitable pipe which extends
therealong.

SUMMARY OF THE INVENTION

[0009] In the present invention, a coordinate system is defined such that
the origin (0,0) of the coordinate system is at the bottom center of the
coordinate system as shown in FIG. 1. Two small rectangular anchor
blocks, with predefined slots, at the two upper corners of the coordinate
system define the width and height of the parabola. These points are
defined as (X1,Y1)=(-width,height), and (X2,Y2)=(width,height). The
equation defining the parabola is f(x)=a.multidot.x.sup.2, where
a=height/width.sup.2. The plot of this equation will produce a parabola
that fits into the coordinate system, and touches the coordinate points
(0,0), (X1,Y1) and (X2,Y2). The two small blocks are used as anchor
points for the ends of the parabola.

[0010] The basic method for building a single reflector utilizes a
clamping system that uses the 1st derivative of the parabola curve at
three specific points to force flexible material to take the shape of a
perfect parabola. The flexible material is calculated to have exactly the
right length to fit the coordinate system. In the past, the shape of a
parabola has only been approximate, or needs a support system that
follows the exact shape of the parabola. This invention provides a method
for constructing a parabolic reflector that has the exact shape of a
parabola, but does not needed to be supported along its entire length to
maintain this shape and is intended to make the construction of a perfect
parabolic reflector less expensive. Described is a basic method of
constructing a perfect parabolic reflector. This method is then used to
construct a six reflector system to concentrate sunlight as well as other
forms of similar parallel incoming radiation from a large rectangular
area to a small rectangular area. This produces a highly concentrated
beam of parallel radiation. A new kind of telescope is described that
also uses the six reflector system.

BRIEF DESCRIPTION OF THE DRAWINGS

[0011] FIG. 1 shows the coordinate system and the calculated points to
form the parabola;

[0012] FIG. 2 shows a parabola of the present invention;

[0013] FIG. 3 shows a two reflector system;

[0014] FIG. 4 further illustrates how the two reflector system works;

[0015] FIG. 5 shows a three dimensional view of a horizontal reflector
system;

[0016] FIG. 6 shows a three dimensional view of a vertical reflector
system;

[0017] FIG. 7 shows a three dimensional view of the combined reflector
systems of FIG. 5 and FIG. 6;

[0018] FIG. 8 depicts how the mounting system works for the three
reflector system;

[0019] FIG. 9 shows a view from the behind and left of a six reflector
system;

[0020] FIG. 10 shows a view from the behind and right of a six reflector
system; and

[0021] FIG. 11 shows how a new kind of telescope functions.

DESCRIPTION OF A PREFERRED EMBODIMENT

[0022] One basic method for Creating a Perfect Parabolic Trough Reflector
can be defined in several steps as set forth below.

[0023] (1) First the height and width of the parabola to be built is
determined.

[0024] (2) A coordinate system (see FIG. 1) is defined, such that the
origin (0,0) is at the bottom dead center of the coordinate system. The
coordinates (-width,height) and (width,height) are defined near the two
upper corners of the coordinate system. As shown in FIG. 1, and
hereinafter (X1,Y1)=(-width,height) and (X2,Y2)=(width,height). The
height of the actual parabola will be Y1 or height and the actual width
of the parabola will be X2-X1 or 2.multidot.width.

[0025] (3) The parabolic equation is defined to be f(x)=a.multidot.x.sup.2
where a=height/width.sup.2. The plot of this equation will produce a
parabola that fits into the coordinate system and touches the coordinate
system at the points (0,0), (X1,Y1), (X2,Y2) and at no other point.

[0026] (4) The points (X3, Y3) and (X4,Y4) are defined to be at the
opposite corners of the two smaller rectangles R1,R2 from the respective
points (X1,Y1) and (X2,Y2) as shown in FIG. 1. The two small rectangles
R1,R2 are the anchor points of the parabola at its end points (X1,Y1) and
(X2,Y2). The third anchor point (0,0) is at the origin. The dimensions of
the coordinate system shown in FIG. 1 are therefore (X4-X3) and (Y4).

[0027] (5) The lines (indicated as S1,S2) at the upper anchor points are a
plot of the lines that have a slope of the 1st derivatives of the
parabola at the points (X1,Y1) and (X2,Y2) and intercept these points.
The 1.sup.st derivative of the parabola f(x)=a.multidot.x.sup.2 is f'
(x)=2.multidot.a.multidot.x.

[0028] (6) The slots (also identified as S1, S2, and defined as the lines
in (5) above) in the support blocks R1, R2 are used to anchor the
parabola. Slots S1,S2 must have the slope of the lines in step (5), touch
the points (X1,Y1) and (X2,Y2) respectively, and extend into the blocks
for a sufficient distance to allow the material used to form the parabola
to be anchored. The width of the slots should match the width of the
reflective material. In addition, the point (0,0) must be anchored to the
bottom dead center of the coordinate system and the 1st derivative of the
curve at this point must be 0. The ends of the parabola in slots S1 and
S2, and the bottom center (0,0) may be anchored, for example, by screws
or clamps.

[0029] (7) The length of the curve of the parabola is calculated as
follows:

f(x)=a.multidot.x.sup.2

dy/dx=2.multidot.a.multidot.x

[0030] To calculate the length of the parabolic curve you start with the
equation for the length of an infinitesimal part of the curve (dl) and
integrate over the length of the curve.

[0031] In the formula, "a" is the coefficient of the parabola defined in
step (3), and "b"=1/(2.multidot.a). The length of the parabolic curve
from the point (X1, Y1) to the point (X2,Y2) is length (X2)-length(X1).
To this is added the length of material that extends into both slots
S1,S2. Both anchor blocks R1,R2 are mirror images of each other so the
slots at both points are of the same length. This means only one kind of
anchor block has to be built. However, mathematically this does not have
to be so, as long as the calculations are done correctly to compensate
for slots of different lengths.

[0032] (8) Once the calculations have been performed, a suitable ridged
support structure is constructed (see FIG. 2, discussed below) to hold
the points (0,0),(X1,Y1),(X2,Y2) of the parabola, and their 1st
derivatives in their proper places. To insure a slope of 0 at the origin
of the coordinate system, the center of the length of the reflective
material from points (X1,Y1) to (X2,Y2) is anchored at the origin by, for
example, a screw, a rivet, or by some other means. The symmetry of the
bending forces of the material will cause the 1st derivative of the
origin to be 0 as required. Each side of the rectangular piece of
reflective material must be supported in this way. The two sides of the
support structure are joined by suitable means to form a parabolic
trough. The material used for the reflector should be of the same
thickness throughout, and must be homogeneous. In addition, the strength
of the material must be strong enough to hold the shape of the parabolic
reflector. Weaker material may be used, but additional support points
along the curve may be required. However, it should not be necessary to
support the material along the entire length of the curve unless it is
chosen to do so.

[0033] FIG. 2 shows an embodiment 10 of a Perfect Parabolic Trough
Reflector. The parabola 11 has its edges 14,15,16,17 in slots S1 and S2
in supports 12 and 13. Edges 14 and 15 are in slot S1 as shown in FIG. 1,
and Edges 16 and 17 are in slot S2 also shown in FIG. 1. The two small
rectangles R1 and R2 found in FIG. 1 are also indicated. The portions of
the parabola 11 extending into supports 12 and 13 are the 1st derivatives
extending from the parabola 11 into the support blocks R1 and R2 as
discussed above in the preferred embodiment. The supports 12 and 13 are
attached to additional supports 18, 19, 20, and 21. The supports
12,13,18,19,20 and 21 along with edges 14,15,16, and 17 form the support
structure, as discussed above, that provides the means for forming and
supporting the parabola.

[0034] The (0,0) point of the parabola in FIG. 2, corresponds to the point
(0,0) in FIG. 1. The points, (X1,Y1) and (X2,Y2) of the parabola are
shown in FIG. 2 and also correspond to the points (X1,Y1) and (X2,Y2) in
FIG. 1.

[0035] The support structure in FIG. 2 is shown as an example. Other
support structures may be used.

[0036] FIG. 2 is representative of what the reflector will really look
like, but it is only an approximation. It is the placement and structure
of the mounting points that is important and not the shape of the support
structure itself which can vary greatly according to design.

[0037] (9) To construct a six reflector system it is first necessary to
describe how to build a two reflector system. If one reflector described
above is a scaled down copy of the other one, and if both reflectors are
arranged to have their focal points coincide, then a trough reflector can
be built that can be used to concentrate parallel radiation such as
sunlight. This arrangement is depicted in FIG. 3. FIG. 3 shows an
incoming light ray striking the larger parabola and being reflected in
the direction of the focus. The angle of reflection is labeled .alpha..
As the ray travels toward the focus, it strikes the second parabola and
is reflected in a direction that is parallel to the incoming light ray. A
slot can be cut in the larger reflector to allow the light ray to exit
through the bottom of the reflector system. The result is a concentrated
rectangular beam of light that is much narrower than the rectangular beam
of light that strikes the larger reflector.

[0038] Mathematically, this can be shown to be true for all angles a that
intercept both parabolas from (h,-w) to (h,w).

[0050] FIG. 3. shows that the slopes of the lines that pass through the
first derivatives of the intercept points of the two parabolas with the
line 1(x) are equal. If two small mirrors depicted as m1 and m2 are
placed at the intercept points of the two parabolas with 1(x) with the
same slope as the first derivatives of the two parabolas at those points,
then they will reflect the incoming ray of light the same way as the
parabolas f(x) and g(x) will. Any incoming light ray that is
perpendicular to the x axis of FIG. 3 will strike the mirror representing
the larger parabola and reflect off of that mirror at the same angle of
incidence that it strikes the mirror according to Snell's law of optics.
The same thing will happen when this reflected ray heading toward the
focus hits the mirror (m2) representing the smaller parabola g(x). What
happens to a ray of light that passes through the two reflectors is
depicted in FIG. 4. The two lines that are perpendicular to the two
parallel mirrors are also parallel, so the alternate interior angles of
these lines are equal. The result is the incoming light ray and the
exiting light rays are parallel.

[0051] Horizontal parabolas arranged as described above will concentrate a
broad parallel beam of sunlight to a narrow beam of the same height as
the reflector. This configuration is shown in FIG. 5. The grey area in
FIG. 5 represents the concentrated sunlight that is coming out of the
back of the horizontal reflectors. The grey area represents an opening in
the rear reflector. FIG. 8 shows this view from the top. The reflector
has been split into two reflectors each with a set of anchor blocks. As
before, the length segments for these parabolic segments can be
calculated using the length(x) formula. The length of the parabolic
segment f1a(x) will equal the length of parabolic segment length f1b(x)
since these segments are mirror images of each other. The length of
segment f1b(x)=length(X8)-length(X6). As before the lengths of the
material extending into the slots are added to the result of this
calculation. The anchor blocks R1 through R7 are calculated to have the
right slopes and the correct positions to support the three reflectors.
S1 through S7 designate the slots as defined in (5) above. P1 through P6
designate the (X,Y) coordinates where the 1.sup.st derivatives of the
parabolic curves are calculated also as defined in (5) above. Note that
the small one piece reflector g1(x) needs a third anchor point where its
1.sup.st derivative is zero just as in the single reflector system. This
sunlight can be further concentrated in the vertical direction by using
another set of parabolic reflectors that are perpendicular to the axis of
the first set of reflectors. The second set of reflectors is shown in
FIG. 6. The second set of three vertical reflectors will be constructed
using the same procedure as the first set of three horizontal reflectors
except the width of the vertical reflectors will match the width of the
back opening of the horizontal reflectors. Again the grey area represents
the beam of sunlight coming out of the back of the larger reflector. The
two three reflector systems can be combined to form a larger six
reflector system by placing the vertical reflectors behind the opening in
the horizontal reflector. This is shown in FIG. 7. The aperture of the
vertical reflectors should match the dimensions of the rear aperture of
the horizontal reflectors, and the apertures should be aligned. One
example of the support structure of this is shown in FIG. 9 and FIG. 10.
Using six reflectors to focus light essentially has the same effect as a
magnifying glass covering the area of the largest reflector. Instead of
focusing light to a small circle or point, the six parabolic reflectors
focus a large rectangular area of sunlight or parallel radiation to a
very small rectangular parallel beam of sunlight or other parallel
radiation. The horizontal reflectors can be interchanged with the
vertical reflectors with the appropriate change in dimensions. This beam
can be used to drive a sterling generator just like the big dish
parabolic reflectors are doing presently, or it can be used to
concentrate light on solar cells. A suitable drive system will have to be
designed to support and rotate the six reflector system to track the
source of radiation. The tracking system will be designed around a
cylindrical coordinate system. It can be used to concentrate microwave
radiation in satellite dishes as well as other applications where
concentration of electromagnetic radiation is desired. It can also be
used for concentrating sound waves. A large reflector of this design can
concentrate sunlight by a factor of thousands of times creating a very
powerful source of energy. Unlike the dish method, there are no molded
spherical surfaces and it is much cheaper to build using the forced
parabola method to build parabolas.

[0052] If a small refracting telescope is placed at the back exit of the
six parabola concentrator, it is possible to build a telescope that has a
much higher power and light gathering capability of the smaller
telescope. The objective of the refracting telescope will take the
parallel light that is coming out of the back of the solar concentrator
and focus it to a point. The solar concentrator will focus an image on
the focal plain of the objective of the smaller telescope. The parallel
light coming out of the back of solar concentrator will be a scaled down
image of the parallel light coming into the front of the concentrator. If
parallel light comes in at an angle from a distant object like a planet
that is off axis of the solar concentrator and the telescope, it focuses
to a point on the focal plane of the objective of the smaller telescope
with little or no distortion. The smaller telescope will function like a
much bigger telescope. The objective of the small telescope will look
like a much larger lens that matches the diameter of the larger solar
concentrator. This means that the lens will have an effective focal
length that is the focal length of the objective of the smaller telescope
multiplied by the ratio of the diameter of the aperture of the solar
concentrator divided by the diameter of the objective of the smaller
telescope. If the small refracting telescope has an objective of 3" and a
focal length of 3 feet, and the solar collector as a square aperture of
30", then the objective of the small telescope will look like a lens with
a diameter of 30", and a effective focal length of 30 feet. Since the
power of a refracting telescope is the ratio of the focal length of the
objective to the focal length of the eyepiece, the power of the
combination of the small refracting telescope and the solar collector
would be 10 times the power of the small refracting telescope alone. A
200 power refracting telescope combined with the solar concentrator
described above would have an effective aperture of 30 inches and a power
of 2,000! FIG. 11 is a basic diagram of the telescope.